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Theorem ablcom 15349
Description: An Abelian group operation is commutative. (Contributed by NM, 26-Aug-2011.)
Hypotheses
Ref Expression
ablcom.b  |-  B  =  ( Base `  G
)
ablcom.p  |-  .+  =  ( +g  `  G )
Assertion
Ref Expression
ablcom  |-  ( ( G  e.  Abel  /\  X  e.  B  /\  Y  e.  B )  ->  ( X  .+  Y )  =  ( Y  .+  X
) )

Proof of Theorem ablcom
StepHypRef Expression
1 ablcmn 15338 . 2  |-  ( G  e.  Abel  ->  G  e. CMnd
)
2 ablcom.b . . 3  |-  B  =  ( Base `  G
)
3 ablcom.p . . 3  |-  .+  =  ( +g  `  G )
42, 3cmncom 15348 . 2  |-  ( ( G  e. CMnd  /\  X  e.  B  /\  Y  e.  B )  ->  ( X  .+  Y )  =  ( Y  .+  X
) )
51, 4syl3an1 1217 1  |-  ( ( G  e.  Abel  /\  X  e.  B  /\  Y  e.  B )  ->  ( X  .+  Y )  =  ( Y  .+  X
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ w3a 936    = wceq 1649    e. wcel 1717   ` cfv 5387  (class class class)co 6013   Basecbs 13389   +g cplusg 13449  CMndccmn 15332   Abelcabel 15333
This theorem is referenced by:  ablinvadd  15354  ablsub2inv  15355  ablsubadd  15356  abladdsub  15359  ablpncan3  15361  ablsub32  15366  eqgabl  15374  subgabl  15375  ablnsg  15382  lsmcomx  15391  divsabl  15400  frgpnabl  15406  ngplcan  18521  r1pid  19942  cnaddcom  29137  toycom  29138  lflsub  29233  lfladdcom  29238
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2361
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2367  df-cleq 2373  df-clel 2376  df-nfc 2505  df-ral 2647  df-rex 2648  df-rab 2651  df-v 2894  df-dif 3259  df-un 3261  df-in 3263  df-ss 3270  df-nul 3565  df-if 3676  df-sn 3756  df-pr 3757  df-op 3759  df-uni 3951  df-br 4147  df-iota 5351  df-fv 5395  df-ov 6016  df-cmn 15334  df-abl 15335
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